Let \(G\) be a disconnected tripartite unicyclic graph on seven edges with two or more connected components. We prove that \(G\) decomposes the complete graph \(K_{n}\) whenever \(n\equiv0,1\pmod{14}\) using labeling techniques.

Let \(G\) be a graph. We introduce the balanced antimagic labeling as an analogue to the antimagic labeling. A \(k\)-total balanced antimagic labelling is a map \(c\colon V (G)\cup E(G) \to \{1,2,\ldots,k\}\) such that: the label classes differ in size by at most one, each vertex \(x\) is assigned the weight \(w(x)={c}(x)+\sum\limits_{x\in e}{c}(e)\), and \(w(x)\neq w(y)\) for \(x\neq y\).

We present several properties of balanced antimagic labeling. We also derive such a labeling for complete graphs and complete bipartite graphs.

For a subgraph \(G\) of the complete graph \(K_n\), a \(G\)-design of order \(n\) is a partition of the edges of \(K_n\) into edge-disjoint copies of \(G.\) For a given graph \(G\), the \(G\)-design spectrum problem asks for which \(n\) a \(G\)-design of order \(n\) exists. This problem has recently been completely solved for every graph \(G\) with less than seven edges, with the lone exception of \(G \cong K_3 \cup 2K_2,\) the disconnected graph consisting of a triangle and two isolated edges. In this article, we solve this problem by proving that a \(K_3 \cup 2K_2\)-design of order \(n\) exists if and only if \(n \equiv 0 \; \textrm{or} \; 1 \pmod{5}\) and \(n\geq 10.\)

A question involving a chess piece called a prince on the \(8\times 8\) chessboard leads to a concept in graph theory involving total domination in the Cartesian products of paths and cycles. A vertex \(u\) in a graph \(G\) totally dominates a vertex \(v\) if \(v\) is adjacent to \(u\). A subset \(S\) of the vertex set of a graph \(G\) is a total dominating set for \(G\) if every vertex of \(G\) is totally dominated by at least one vertex of \(S\). If \(S\) is a total dominating set of a graph \(G\), then \(S(v)\) denotes the number of vertices in \(S\) that totally dominate \(v\). A total dominating set \(S\) in a graph \(G\) is called a proper total dominating set if \(S(u) \ne S(v)\) for every two adjacent vertices \(u\) and \(v\) of \(G\). It is shown that \(C_n \ \Box \ K_2\) possesses a proper total dominating set if and only if \(n\ge 4\) is even and the graph \(C_n \ \Box \ P_m\) possesses a proper total dominating set for every even integer \(n \ge 4\) and every integer \(m \ge 3\). Furthermore, \(C_3 \ \Box \ P_m\) possesses a proper total dominating set if and only if \(m = 3\). If \(n\ge 5\) is an odd integer and \(m\equiv 3 \pmod 4\), then \(C_n \ \Box \ P_m\) has a proper total dominating set. If at least one of \(n\) and \(m\) is even, then \(C_n \ \Box \ C_m\) has a proper total dominating set. The graphs \(C_n \ \Box \ C_m\) are further studied when both \(n\) and \(m\) are odd. Other results and questions are also presented.

For a graph \(F\) and a positive integer \(t\), the vertex-disjoint Ramsey number \(VR_t(F)\) is the minimum positive integer \(n\) such that every red-blue coloring of the edges of the complete graph \(K_n\) of order \(n\) results in \(t\) pairwise vertex-disjoint monochromatic copies of subgraphs isomorphic to \(F\), while the edge-disjoint Ramsey number \(ER_t(F)\) is the corresponding number for edge-disjoint subgraphs. These numbers have been investigated for the three connected graphs \(K_3\), \(P_4\) and \(K_{1, 3}\) of size 3. For two vertex-disjoint graphs \(G\) and \(H\), let \(G+H\) denote the union of \(G\) and \(H\). Here we study these numbers for the two disconnected graphs \(3K_2\) and \(P_3+P_2\) of size 3. It is shown that \(VR_t(3K_2)= 6t+2\) and \(VR_t(P_3+P_2)= 5t+1\) for every positive integer \(t\). The numbers \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) are determined for \(t \le 4\) and bounds are established for \(ER_t(3K_2)\) and \(ER_t(P_3+P_2)\) when \(t \ge 5\). Other results and problems are presented as well.

Every red-blue coloring of the edges of a graph \(G\) results in a sequence \(G_1\), \(G_2\), \(\ldots\), \(G_{\ell}\) of pairwise edge-disjoint monochromatic subgraphs \(G_i\) (\(1 \le i \le \ell\)) of size \(i\) such that \(G_i\) is isomorphic to a subgraph of \(G_{i+1}\) for \(1 \le i \le \ell-1\). Such a sequence is called a Ramsey chain in \(G\) and \(AR_c(G)\) is the maximum length of a Ramsey chain in \(G\) with respect to a red-blue coloring \(c\). The Ramsey index \(AR(G)\) of \(G\) is the minimum value of \(AR_c(G)\) among all red-blue colorings \(c\) of \(G\). Several results giving the Ramsey indexes of graphs are surveyed. A galaxy is a graph each of whose components is a star. Results and conjectures on Ramsey indexes of galaxies are presented.

Given a network modeled as a graph, a detection system is a subset of vertices equipped with “detectors” that can uniquely identify an “intruder” anywhere in the graph. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). In an OLD set, each vertex has a unique, non-empty set of detectors in its open neighborhood; meanwhile, in an IC, each vertex has a unique, non-empty set of detectors in its closed neighborhood. We explore one of their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector retains the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king grid, and presents the proof for the bounds on their minimum densities; \(\left[\frac{3}{10}, \frac{1}{3}\right]\) for RED:OLD sets and \(\left[\frac{3}{11}, \frac{1}{3}\right]\) for RED:ICs.

Exploring the vulnerability of any real-life network helps designers understand how strongly components or elements of the network are connected and how well they can function if there is any disruption. Any chemical structure can also be considered as a network in which the atoms correspond to the vertices, and the chemical bonds between the atoms correspond to the edges. Let \(G=(V, E)\) represent any simple graph with vertex set \(V\) and edge set \(E\). The vulnerability measure used in this paper is the paired domination integrity, defined as the minimum of the sum of any paired dominating set \(S\) of a graph \(G\) and the order of the largest component in the induced subgraph of \(V-S\). The minimum is found by considering all possible paired dominating sets of \(G\). In this paper, we obtain the paired domination integrity of the comb product of paths and cycles. In addition, we extend the study of graph vulnerability to chemical structures.

Let \(k, b, n\) be positive integers such that \(b\geq 2\). Denote by \(S(k,b,n)\) the numerical semigroup generated by \(\left\{b^{k+n+i}+\frac{b^{n+i}-1}{b-1}\mid i\in\mathbb{N}\right\}\). In this paper, we give formulas for computing the embedding dimension and the Frobenius number of \(S(k,b,n)\).

Given a connected graph \(G=(V,E)\) of order \(n\ge 2\) and two distinct vertices \(u,v\in V(G)\), consider two operations on \(G\): the \(k\)-multisubdivision and the \(k\)-path addition. Let \(msd_{\gamma_c}(G)\) and \(pa_{\gamma_c}(G)\) denote, respectively, the connected domination multisubdivision and path addition numbers of \(G\). In both operations, \(k\) represents the number of vertices added to \(V(G)\), resulting in a new graph denoted by \(G_{u,v,k}\). We prove that \(\gamma_c(G) \le \gamma_c(G_{u,v,k})\) for \(k = msd_{\gamma_c}(G) \in \{1,2,3\}\) in the case of \(k\)-multisubdivision, where \(uv \in E(G)\). Additionally, we show that \(\gamma_c(G) – 2 \le \gamma_c(G_{u,v,k})\) for \(k = pa_{\gamma_c}(G) \in \{0,1,2,3\}\) in the case of \(k\)-path addition, where \(uv \notin E(G)\), and provide both necessary and sufficient conditions under which these inequalities hold.